Spectral Theory in Hilbert Space

11. STURM-LIOUVILLE EQUATIONS 71
Proof. Suppose u is in the domain of T , i.e., T u = v for some
v ∈ L 2 (I), and let ˜v = v − iu, so that u = Ri˜v. If e is an eigenfunction
of T with eigenvalue λ we have T e = λe or (T + i)e = (λ + i)e so that
R−ie = e/(λ + i). It follows that 〈u, e〉 e = 〈Ri˜v, e〉 e = 〈˜v, R−ie〉 e =
1
λ−i 〈˜v, e〉 e = 〈˜v, e〉Rie. If sNu denotes the N:th partial sum of the
Fourier series for u it follows that sNu = RisN ˜v, where sN ˜v is the N:th
partial sum for ˜v. Since sN ˜v → ˜v in L 2 (I), it follows from Theorem 11.1
and the remark after it that sNu → u in C 1 (K), for any compact
subinterval K of I.
The convergence is actually even better than the corollary shows,
since it is absolute and uniform (see Exercise 11.2).
Example 11.5. Consider the equation −u ′′ = λu, first in L2 (−π, π),
with periodic boundary conditions u(−π) = u(π), u ′ (−π) = u ′ (π). The
general solution is u(x) = A cos( √ λ x) + B sin( √ λ x), where A, B are
constants. The boundary conditions may be viewed as linear equations
for determining the constants A and B, and if there is going to be a
non-trivial solution, the determinant must vanish. The determinant is
0 2 sin(
√ λ π)
−2 sin( √
λ π) 0 = 4 sin2 ( √ λ π)
so that λ = k 2 , where k ∈ N. For each eigenvalue k 2 > 0 we have
two linearly independent eigenfunctions cos(kx) and sin(kx). For the
eigenvalue 0 the eigenfunction is 1 √ . These functions are orthonormal
2
if we use the scalar product 〈u, v〉 = 1
π
uv (check!). We obtain the
π −π
classical (real) Fourier series f(x) = a0
2 + ∞ k=1 (ak cos kx + bk sin kx),
where a0 = 〈f, 1〉, ak = 〈f(x), cos kx〉 for k > 0, and bk = 〈f(x), sin kx〉.
In this case Corollary 11.4 states that the series for u as well as that
for u ′ converge uniformly if u is continuously differentiable with an
absolutely continuous derivative such that u ′′ ∈ L 2 (−π, π).
Now consider the same equation in L 2 (0, π), with separated boundary
conditions u(0) = 0 and u(π) = 0. Applying this to the general
solution we obtain first B = 0 and then A sin √ λ π) = 0, so
a non-trivial solution exists only if λ = k 2 for a positive integer k.
Thus the eigenfunctions are sin x, sin 2x, . . . . These are orthonormal
if the scalar product used is 〈u, v〉 = 2
π
uv. We obtain a sine se-
π 0
ries f(x) = ∞ k=1 bk sin(kx), where bk = 〈f(x), sin kx〉. This is the
series expansion relevant to the vibrating string problem discussed in
Chapter 0 (if the length of the string is π).
Finally, consider the same equation, still in L2 (0, π), but now with
separated boundary conditions u ′ (0) = 0 and u ′ (π) = 0. Applying this
to the general solution we obtain first A = 0 and then B sin( √ λ π) = 0,
so a non-trivial solution requires λ = k2 for a non-negative integer k.
Thus the eigenfunctions are 1
√ 2 , cos x, cos 2x, . . . . These are orthonormal
with the same scalar product as in the previous example. We